6.1 Solving Inequalities in One Variables 
          
          
         Objectives:   
             •  Learn how to graph linear inequalities in one variable 
             •  Learn how to solve linear inequalities in one variable 
          
          
         Graphing Inequalities: 
          
         Phrase Inequality Graph 
         Less than                                           x < 2 
                                                                                      –5 –4 –3 –2 –1 010      2345x 
         Greater than                                        x > 5 
                                                                                      –2 –1 010     2345678x 
         Less than or equal to                               x ≤ 3 
                                                                                      –2 –1 010     2345678x 
         Greater than or equal to                           x ≥ –2 
                                                                                      –5 –4 –3 –2 –1 010      2345x 
          
                                                     - open circle for >, < 
                                                     - closed circle for ≥, ≤ 
          
                                                         T ≥ 80° 
         Summer temperature in Phoenix, 
               00T20406080100120
                                                                        
          
         Average snowfall is less than 1” 
                –3      –2      –1       010              23
                                                                        
          
         Can it go below zero? 
          
         Mr. Noyes, Akimel A-al Middle School                         1               Heath Algebra 1 - An Integrated Approach  
        6.1 Solving Inequalities in One Variables 
         
         
        Solving linear inequalities in one variable: 
                Graph on a number line not coordinate plane (Cartesian plane) 
         
         
        We solve inequalities in one variable much like we solve equations in one variable.  
        We need to isolate the variable on one side by using transformations. 
         
         
        œœ However œœ  If you multiply or divide by a negative number - change the 
                                  sign (reverse the direction) of the inequality 
         
                              2x + 3 = 4                                  2x + 3 > 4    
                                    –3    –3                                    –3   –3 
                                    2x = 1                                      2x > 1 
                                    2     2                                     2      2 
                                     x =  1                                       x >  1  
                                          2                                         2
                     –1              010                       –1             010
                                                                                                   
                           Graph (plot point)                   Graph (use open circle) 
         
                                  3 + x ≥  9                             x – 8 < –10 
                                –3          –3                              + 8   + 8 
                                       x  ≥  6                                 x <  – 2 
                                                                     
                      012340             5678  –5 –4 –3 –2 –1 0120                           3 
                      Graph (use closed circle)                 Graph (use open circle) 
         
         
         
        Mr. Noyes, Akimel A-al Middle School                    2             Heath Algebra 1 - An Integrated Approach  
        6.1 Solving Inequalities in One Variables 
         
         
        Solve and Graph! 
                            – x – 8 > –17                            x + 3 ≤ 2 (x – 4) 
                                  + 8     + 8                        x + 3 ≤ 2x – 8 
                            – x       >  – 9                      – x           –x      
                             x       >  – 9 (–1) 
                    (–1) –                                                 3 ≤ x – 8 
                                    x  <  9                            + 8       + 8 
                                                                         11 ≤ x 
                                                              re–write x ≥ 11 
                      00x12345678910                             5 6 7 8 9101112131415 x
                                                                                                       
                        Graph (use open circle)                   Graph (use closed circle) 
          
         
         
         
        In Oswego, NY, the temperature in January may not exceed 0° C.  Write an inequality 
        that describes the temperature T for the month and graph it. 
         
                                                               T ≤ 0 
                                  –10–9 –8 –7 –6 –5 –4 –3 –2 –1 010     2345678910x 
         
         
         
         
         
         
         
        Mr. Noyes, Akimel A-al Middle School                      3              Heath Algebra 1 - An Integrated Approach  
       6.2 Problem Solving Using Inequalities 
        
        
       Objectives:   
           •  Learn how to write and use a linear inequality as a model for a real-life situation 
        
        
       Example 1: 
       You have $32.14 to spend, but sales tax is 6%, so how much can you spend so that 
       you still have enough money to pay the sales tax? 
        
              Verbal Model: 
                     cost + sales tax ≤ Total money to spend 
               
              Labels
                      : 
                     c = cost of items purchased 
               
              Algebraic Inequality: 
                     c + 0.06c ≤ 32.14 
                             1.06c ≤ 32.14 
                              1.06     1.06 
                                    c ≤ 30.32 
        
       Example 2: 
       If you go into business selling model cars for $18 and costs you $2,000 to set-up the 
       business and then $13 per car to produce them, how many cars must you sell to 
       realize a profit? 
              Verbal Model: 
                     # of cars · 18 ≥ # of cars · 13 + set-up cost 
       Mr. Noyes, Akimel A-al Middle School               1            Heath Algebra 1 - An Integrated Approach